Weakly Birkhoff recurrent switching signals, almost sure and partial stability of linear switched dynamical systems

作者：Dai Xiongping^*

来源：Journal of Differential Equations, 2011, 250(9): 3584-3629.

DOI：10.1016/j.jde.2011.01.029

• Linear switched system
• Stability
• Recurrence
• Rotation number
• Switching indicator
• Exponential stabilizability
• Quasi-additive ergodic theorem

Let {A(I), ... , A(K)) subset of C-dxd be arbitrary K matrices, where K and d both >= 2. For any 0 < Delta < infinity, we denote by L-Delta(pc)(R+. K) the set of all switching sequences u = (lambda., t.): N -> (1, ... , K} x R+ satisfying t(j) - t(j-i) <= Delta and
0=: t(0) < t(1) < ... < t(j-1) < t(j) < ... with t(j) -> +infinity
Differently from the classical weak-* topology and L-1-norm, we equip L-Delta(pc)(R+. K) with the topology so that the "one-sided Markov-type shift" v(+) :L-Delta(pc)(R+, K) L-Delta(pc)(R+, K), defined by
u = (lambda(j), t(j))(j=1)(+infinity) -> v(+)(u) = (lambda(j+1), t(j+1) - t(1))(j=1)(+infinity),
is continuous, which is different from and simpler than the classical continuous-time "translation". We study the stability of the linear switched dynamics (A):
<(x)over dot>(t) = A(u(t))x(t), x(0) is an element of C-d and t> 0
where u(t) lambda(j) if t(j-1) < t <= t(j), for any u is an element of L-Delta(pc)(R+, K). By introducing the concept "weakly Birkhoff recurrent switching signal", we show that if, under some norm parallel to . parallel to, the principal matrix Phi(u)(t) of (A) satisfies parallel to Phi(u)(t)parallel to <= 1 for all u is an element of L-Delta(pc)(R+, K) and t > 0, then for any 0+-ergodic probability P on u is an element of L-Delta(pc)(R+, K). either
lim 1/j log parallel to Phi(u)(t(j))parallel to < 0 for P-a.s. u = (lambda(j), t(j))(j=1)(+infinity); j ->infinity
or
parallel to Phi(vj+(u)) (t(j+k) - t(j))parallel to = 1 (sic), j >= 0 for P-a.s. u = (lambda(j), t(j))(j=1)(+infinity).
Some applications are presented. including: (i) equivalence of various stabilities; (ii) almost sure exponential stability of periodically switched stable systems; (iii) partial stability; and (iv) how to approach arbitrarily the stable manifold by that of periodically switched signals and how to select a stable switching signal for any initial data.

• 出版日期2011-5-1
• 单位南京大学